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Theorem sstpr 3502
Description: The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
sstpr ((((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) ((A = {𝐷} A = {B, 𝐷}) (A = {𝐶, 𝐷} A = {B, 𝐶, 𝐷}))) → A ⊆ {B, 𝐶, 𝐷})

Proof of Theorem sstpr
StepHypRef Expression
1 ssprr 3501 . . 3 (((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) → A ⊆ {B, 𝐶})
2 prsstp12 3491 . . 3 {B, 𝐶} ⊆ {B, 𝐶, 𝐷}
31, 2syl6ss 2934 . 2 (((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) → A ⊆ {B, 𝐶, 𝐷})
4 snsstp3 3490 . . . . 5 {𝐷} ⊆ {B, 𝐶, 𝐷}
5 sseq1 2943 . . . . 5 (A = {𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {𝐷} ⊆ {B, 𝐶, 𝐷}))
64, 5mpbiri 157 . . . 4 (A = {𝐷} → A ⊆ {B, 𝐶, 𝐷})
7 prsstp13 3492 . . . . 5 {B, 𝐷} ⊆ {B, 𝐶, 𝐷}
8 sseq1 2943 . . . . 5 (A = {B, 𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {B, 𝐷} ⊆ {B, 𝐶, 𝐷}))
97, 8mpbiri 157 . . . 4 (A = {B, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
106, 9jaoi 623 . . 3 ((A = {𝐷} A = {B, 𝐷}) → A ⊆ {B, 𝐶, 𝐷})
11 prsstp23 3493 . . . . 5 {𝐶, 𝐷} ⊆ {B, 𝐶, 𝐷}
12 sseq1 2943 . . . . 5 (A = {𝐶, 𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {𝐶, 𝐷} ⊆ {B, 𝐶, 𝐷}))
1311, 12mpbiri 157 . . . 4 (A = {𝐶, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
14 eqimss 2974 . . . 4 (A = {B, 𝐶, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
1513, 14jaoi 623 . . 3 ((A = {𝐶, 𝐷} A = {B, 𝐶, 𝐷}) → A ⊆ {B, 𝐶, 𝐷})
1610, 15jaoi 623 . 2 (((A = {𝐷} A = {B, 𝐷}) (A = {𝐶, 𝐷} A = {B, 𝐶, 𝐷})) → A ⊆ {B, 𝐶, 𝐷})
173, 16jaoi 623 1 ((((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) ((A = {𝐷} A = {B, 𝐷}) (A = {𝐶, 𝐷} A = {B, 𝐶, 𝐷}))) → A ⊆ {B, 𝐶, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616   = wceq 1228  wss 2894  c0 3201  {csn 3350  {cpr 3351  {ctp 3352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3or 874  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-tp 3358
This theorem is referenced by:  pwtpss  3551
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